My question is:
How many different equivalence relations can we define on the set $A = \{x,y,z\}$?
I know that an equivalence relation is a relation that is symmetric, reflexive, and transitive, so how many I go about considering these possible relations?
I am particularly curious about this relation: $\{(x,x), (y,y), (z,z)\}$
Is it one possibility?
Best Answer
An equivalence relation is defined by its equivalence classes. Given an equivalence relation, its equivalence classes constitute a partition of the set $A$.
Hence, an easy way to count the number of equivalence relations is to count the number of ways in which $A$ can be partitioned. This is provided by the Bell number $B_3 = 5$.